Method for the detection of an electromagnetic signal by an antenna array, and device implementing said method

ABSTRACT

A method for detecting an electromagnetic signal comprises: applying to the received electromagnetic signal a plurality of time-frequency transforms, for each time/frequency cell of a given set of cells, calculating the energy of the vector made up of the spectra over all of the antenna elements, applying the following nonlinear function T to the result of the energy calculation: if the norm of the energy is below a first predetermined threshold s, the result of the function T is zero, if the norm of the energy is above or equal to the first threshold s, the result of the function T is equal to the norm of the energy minus the value of the first threshold s, integrating, over the set of time/frequency cells, the result of the nonlinear function T, comparing the result of the integration to a second predetermined threshold, to detect the presence of the signal.

The invention relates to the field of the detection of wirelessemissions originating, notably, from radars or from telecommunicationsystems, and received by an antenna array with space diversity.

The invention relates more specifically to a method for detecting anelectromagnetic signal by an antenna array, notably a lacunary antennaarray, and a device for implementing said method.

One problem to be solved in the field of the detection ofelectromagnetic signals lies in the fact that the type of signalintercepted is not a priori known, particularly its frequency bandwidth,the type of modulation used or, more generally, any parameter associatedwith the waveform of the signal.

The known detection methods are generally constructed on the a prioriknowledge of the form of the signal and use a filter adapted accordingto this knowledge.

However, it is not possible to implement filters adapted to all types ofsignals expected.

Two major types of receivers have hitherto been considered to provide awatch over a very wide frequency band: the receivers permanentlycovering the band to be watched, which are adapted to detect only thesignals of high power, and the narrowband receivers, which do not makeit possible to instantaneously cover the total band, but whose functionis to detect signals of lower power and which allow for finer analysesof the signal.

The present invention falls within the scope of the narrowbandreceivers.

The traditional methods for detecting electromagnetic signals arenotably based on the following preliminary steps.

The reception of signals is done through an antenna array with spacediversity, or interferometric array, and the demodulation of the signalis performed by the same local oscillator for all the sensors of thearray. The signal is then sampled, on each reception channel, in real orcomplex form, then one or more banks of filters are applied, for exampleby weighted discrete Fourier transform. In other words, a number oftemporally overlapped discrete Fourier transforms are applied in orderto produce an average adaptation to the band of the signals of interest.At the end of this operation, called time-frequency analysis, the signalis transformed into a time-frequency grid broken down intotime-frequency cells, each cell containing the result of a discreteFourier transform for a given time interval and a given frequencyinterval.

One known detection method consists in comparing the power of thesignal, in each time-frequency cell, to a given detection threshold.However, this cell-by-cell decision-making is not optimal when thesignal is spread in time and/or in frequency.

There is therefore a problem to be solved in adapting to the spread ofthe signal over a plurality of time-frequency cells.

Also known are solutions based on the formation of a single channel fromthe signals received by the plurality of antenna elements of the array,but these solutions are efficient only for regular arrays, in otherwords arrays for which the distance between two antenna elements isregular.

There is therefore also a problem to be solved in designing a detectionmethod adapted to a lacunary array, that is to say an array for whichthe spacing between two elements is not regular.

The invention proposes a method and a system for detectingelectromagnetic signals which does not require any a priori knowledge ofthe type of signal and which can be implemented through a lacunaryantenna array.

The invention is suited to both antenna arrays with mono-polarizationand with bipolarization.

Thus, the subject of the invention is a method for detecting anelectromagnetic signal, called signal of interest, likely to becontained in an electromagnetic signal received by an antenna arraycomprising a plurality of antenna elements, said method comprising thefollowing steps:

applying to said electromagnetic signal received by each antenna elementa plurality of time-frequency transforms in order to obtain arepresentation of said signal in the form of a plurality oftime-frequency cells each containing the spectrum of said signal for agiven frequency interval and a given time interval,

for each time/frequency cell of a given set of cells,

-   -   calculating the energy of the vector made up of the spectra over        all of the antenna elements,    -   applying the following nonlinear function T to the result of the        preceding energy calculation, so as to cancel the time/frequency        cells containing substantially only noise:        -   if the norm of the energy of the vector of the spectra is            below a first predetermined threshold s, the result of the            function T is zero,        -   if the norm of the energy of the vector of the spectra is            above or equal to said first threshold s, the result of the            function T is equal to the norm of the energy of the vector            of the spectra minus the value of said first threshold s,

integrating, over said set of time/frequency cells, the result of saidnonlinear function T,

comparing the result of the integration to a second predeterminedthreshold, called detection threshold S_(det), to detect the presence ofthe signal of interest.

According to a particular aspect of the invention, said antenna elementsare mono-polarized.

According to another particular aspect of the invention, said antennaelements are bipolarized.

In the case where said antenna elements are bipolarized, the energycalculation step can be performed on the vector made up of the spectrafor each of the polarizations over all of the antenna elements.

In the case where said antenna elements are bipolarized, the methodaccording to the invention can further comprise the following steps:

calculating the covariance matrix between the plurality of spectraassociated with the plurality of antenna elements configured accordingto a first polarization and the plurality of spectra associated with theplurality of antenna elements configured according to a secondpolarization,

calculating the eigenvector associated with the greatest eigenvalue ofsaid covariance matrix,

in the energy calculation step, replacing the spectrum with its scalarproduct with said eigenvector.

In the case where said antenna elements are bipolarized, the steps ofcalculation of the energy of the spectra, of application of thenonlinear function T and of integration, over a plurality oftime/frequency cells, of the result of said nonlinear function T, can beperformed separately for each polarization of said antenna elements,said method further comprising an additional step of determination ofthe maximum of the results of integration over the two polarizations,said maximum being compared to said second detection threshold.

According to another particular aspect of the invention, said firstthreshold s is determined by searching for the intersection between thex-axis and the asymptote at infinity of the logarithm of the likelihoodratio defined as the quotient of the probability densities in thehypothesis in which the signal of interest is present in atime/frequency cell and in the hypothesis in which the signal ofinterest is absent in a time/frequency cell.

The first threshold s can be calculated as a function of a givensignal-to-noise ratio and of a parameter q representative of theprobability of presence of the signal of interest in a time/frequencycell.

Said parameter q can be chosen from a range lying between 0.1 and 1.

According to another particular aspect of the invention, said seconddetection threshold S_(det) is configured to observe a given probabilityof false alarm.

According to another particular aspect of the invention, said antennaarray is lacunary.

Another subject of the invention is a device for the detection of anelectromagnetic signal, called signal of interest, likely to becontained in an electromagnetic signal, said device comprising anantenna array formed by a plurality of antenna elements and meansconfigured to implement the method according to the invention.

Other features and advantages of the present invention will become moreapparent on reading the following description in relation to theattached drawings which represent:

FIG. 1, a flow diagram of the steps of implementation of the method fordetecting electromagnetic signals according to the invention,

FIG. 2a , a block diagram of a device for the detection ofelectromagnetic signals for a mono-polarized lacunary array according toa first embodiment,

FIG. 2b , a block diagram of a device for the detection ofelectromagnetic signals for a mono-polarized lacunary array, accordingto a variant of the first embodiment,

FIG. 3a , a block diagram of a device for the detection ofelectromagnetic signals for a bipolarized lacunary array, according to afirst and a second embodiment,

FIG. 3b , a block diagram of a device for the detection ofelectromagnetic signals for a bipolarized lacunary array, according to athird embodiment.

DETECTION ON ANTENNA ARRAY WITH MONO-POLARIZATION

The invention is now described according to a first embodiment whichrelates antenna arrays with mono-polarization, in other words, thearrays which are made up of antenna elements polarized according to asingle polarization.

The method according to the invention uses the outputs of thetime-frequency analysis processing, in other words the time-frequencycells, to decide on the presence or the absence of a signal, in thepresence of thermal noise which is modeled as a centered complex randomGaussian signal of symmetrical spectral density N₀/2 for each of itsreal and imaginary components which are mutually independent.

To construct the detector, the statistical decision theory is used,described notably in the work “Testing Statistical Hypothesis, E. L.Lehmann, J. P. Romano, Springer 2005” which amounts to testing thevalidity of one of the two models of the signal received, in other wordsthe presence, denoted H₁, or the absence, denoted H₀, of the usefulsignal, so as to optimize the Neyman-Pearson criterion which consists inmaximizing the probability of detection, subject to the constraint thatthe probability of false alarm is fixed.

For a given discrete Fourier transform size, the hypothesis H₁ will betested against H₀ in a time-frequency window made up of the outputs ofseveral consecutive discrete Fourier transforms limited to a given bandand to a given time interval.

The outputs in “time” and in “frequency” extend vectorially, each of thecomponents originating from one of the P sensors of the antenna array.

To avoid unnecessarily complicating the notations, the duly formed“time-frequency” cell will be designated by the index n. Since theuseful signal is narrowband, it is always written vectorially in thefollowing form for the cell n: Us_(n) in which s_(n) is the component ofthe demodulated useful signal, in other words its complex envelope,projected by virtue of the discrete Fourier transform operation. U is afixed vector for the same threat whose components reflect theinterferometric phase shifts for an interferometric antenna array.

To complete the modeling of the useful signal, information has to begiven on s_(n), and that represents a difficulty since, in the listeningdomain, the signal is by definition unknown. Now, it is precisely byexploiting a realistic model that it is possible to make progress inefficiency. Given the filtering effects, after the discrete Fouriertransforms, the useful signal present in the time-frequency cell n,s_(n), is modeled by a sample of a centered complex random Gaussianvariable with independent components of common variance σ′². Moreover,the different values of s_(n) are considered as independent. This allreflects a model with little constraint which makes no assumption ofphase or amplitude continuity so as to be valid for all the possiblemodulations.

The following model is obtained for the received signal in the casewhere the useful signal is present (H₁) and where it is absent (H₀):

$\left\{ {{\begin{matrix}{H_{1}\text{:}} & {X_{n} = {{Us}_{n} + W_{n}}} \\{H_{0}\text{:}} & {X_{n} = W_{n}}\end{matrix};{n = 1}},2,{\ldots \mspace{14mu} {N.}}} \right.$

in which W_(n) is the digital white noise of covariance matrix 2σ²I withI being the identity matrix of size P×P. It is assumed that the power ofthe noise 2σ² is known or can be estimated elsewhere. With no loss ofgenerality, it will always be assumed hereinbelow that U is normed,which amounts to changing the value of σ′² as required.

To complete the above model, in the hypothesis H₁, it will be added thats_(n) is present in the above form with the probability q and absentwith the probability 1−q, and in a way that is independent of n.

This model involves unknown parameters: the directions (θ,φ) in U,σ′²,q.

In H₁, conditionally on the directions (θ,φ), and on the fact that s_(n)is different from zero.

E(X_(n)X*_(n)|U)=2σ′²UU*+2σ²I in which I denotes the identity matrix ofdimension PxP in which P is the number of antennas, and in which thenotation X* denotes the conjugate transpose of X.

If all of the directions of arrivals of the targets are considered(which amounts to saying that the aim is to make a good on averagedetector for all the directions of arrival of the targets), the averageof the terms UU* is proportional to I.

The non-conditional covariance is deduced therefrom in H₁:E(X_(n)X*_(n))=(2σ′²+2σ²)I. The term 2σ′² “absorbs” the coefficient ofproportionality between UU*and I: to avoid complicating the notations,the term 2σ′² is retained rather than introducing a term 2σ″².

And, in H₀, E(X_(n)X*_(n))=2σ²I

In the hypothesis H₁, the sample X_(n) has the probability density

${p_{1}\left( X_{n} \right)} = {{\frac{q}{{\pi^{P}\left( {2\left( {\sigma^{\prime 2} + \sigma^{2}} \right)} \right)}^{P}} \cdot {\exp \left( \frac{{- X_{n}^{*}}X_{n}}{2\left( {\sigma^{\prime 2} + \sigma^{2}} \right)} \right)}} + {\frac{1 - q}{{\pi^{P}\left( {2\sigma^{2}} \right)}^{P}}\left( \frac{{- X_{n}^{*}}X_{n}}{2\sigma^{2}} \right)}}$

The probability density of the measurement X_(n) in the hypothesis inwhich there is absence of signal is:

${p_{0}\left( X_{n} \right)} = {\frac{1}{{\pi^{P}\left( {2\sigma^{2}} \right)}^{P}} \cdot {\exp \left( \frac{{- X_{n}^{*}}X_{n}}{2\sigma^{2}} \right)}}$

The likelihood ratio L(X_(n)) is the quotient of the probabilitydensities in the hypotheses H₁ and H₀.

${L\left( X_{n} \right)} = {\frac{p_{1}\left( X_{n} \right)}{p_{0}\left( X_{n} \right)} = {{\frac{q \cdot \sigma^{2\; P}}{\left( {\sigma^{\prime 2}\sigma^{2}} \right)^{P}} \cdot {\exp \left( \frac{\sigma^{\prime 2}X_{n}^{*}X_{n}}{2{\sigma^{2}\left( {\sigma^{\prime 2}\sigma^{2}} \right)}} \right)}} + \left( {1 - q} \right)}}$

To determine the detection test according to the invention, thelogarithm of the likelihood ratio is calculated for all the measurementsX_(n); n=1, 2, . . . , N:

${L\left( {X_{1},X_{2},\ldots \mspace{14mu},{X_{N};q},\sigma^{\prime 2}} \right)} = {{{\sum\limits_{n = 1}^{N}{\ln \mspace{11mu} {p\left( {q,X_{n}} \right)}}} - {\ln \mspace{11mu} {p\left( {0,X_{n}} \right)}}} = {\sum\limits_{n = 1}^{N}{{\ln \left( {{q \cdot a^{P} \cdot {\exp \left( {{bX}_{n}^{*}X_{n}} \right)}} + \left( {1 - q} \right)} \right)}\mspace{14mu} {with}}}}$$\mspace{20mu} {a = {{\frac{\sigma^{2}}{\left( {\sigma^{\prime 2} + \sigma^{2}} \right)}\mspace{14mu} {and}\mspace{14mu} b} = \frac{\sigma^{\prime 2}}{2 \cdot {\sigma^{2}\left( {\sigma^{\prime 2} + \sigma^{2}} \right)}}}}$

by using the independence of the measurements Xn.

As this function depends on two unknown parameters q and σ′², it is notpossible to use it as such in the detection between “simple hypotheses”.According to the invention, the operation of the detector is thenoptimized in the vicinity of the signal-to-noise ratio K; which amountsto setting σ′²=K σ². As for the parameter q, it can be set if there isan average idea of the time-frequency occupancy of the useful signal inthe detection window.

The comparison of the likelihood ratio L(X₁, X₂, . . . , X_(N); q,σ′²)to a threshold (optimal detector in the sense of the Neyman-Pearsoncriterion for the values of q and σ′² chosen), is equivalent tocomparing the function Tq to a threshold:

${\sum\limits_{n = 1}^{N}{T_{q}\left( {X_{n}^{*}X_{n}} \right)}} = {{\sum\limits_{n = 1}^{N}{\alpha \; {L_{n}\left( {X_{n}^{*}X_{n}} \right)}}} + \beta}$

in which α>0 and β real.

The aim is to have the asymptotic behavior of the detector be that ofthe quadratic detector, which is reflected by the relationship

${\lim\limits_{{X_{n}}^{2}->\infty}\frac{T_{q}\left( {X_{n}^{*}X_{n}} \right)}{X_{n}^{*}X_{n}}} = 1.$

Furthermore, the aim is that, regardless of q, Tq(0)=0 (which means thatwhen the measured signal is zero, the criterion must be zero).

To satisfy these constraints, it is essential to choose: α=1/b andβ=−α·ln(q·α^(P)+1−q)

The function T_(q) is therefore written:

$\begin{matrix}{{T_{q}\left( {X_{n}^{*}X_{n}} \right)} = {{\frac{1}{b} \cdot \ln}\frac{{q \cdot a^{P} \cdot {\exp \left( {{bX}_{n}^{*}X_{n}} \right)}} + \left( {1 - q} \right)}{{q \cdot a^{P}} + 1 - q}}} & (1)\end{matrix}$

The function T_(q) is a positive function defined on the positive realnumbers.

When q<1, T_(q)(u) tends toward 0 when u tends toward 0, and has forasymptote the straight line u-u₀ when u tends toward +∞, with

$\begin{matrix}{u_{0} = {{- \frac{1}{b}}\ln {\frac{q \cdot a^{P}}{{q \cdot a^{P}} + 1 - q}.}}} & (2)\end{matrix}$

For q=1, exactly T_(q)(u)=u is found.

It is possible to liken the behavior of T_(q) to two straight lines (theasymptote of T_(q) as infinity, and the x-axis), and to retain only thisbehavior to define the detector h according to the invention, whichamounts to performing:

Σ_(n=1) ^(N) h(∥X _(n)∥²)> or <threshold

in which (u)=0 for 0≦u≦u₀ and h(u)=u−u₀ for u>u₀, with u₀>0 for q<1 andu₀=0 for q=1.

This detector is therefore made up of a spatially incoherent integration(∥X_(n)∥²), followed by a nonlinearity (function h) then by anincoherent integration over a time/frequency window (Σ_(n=1) ^(N)).

Detection on Antenna Array with Bipolarization

The invention is now described according to a second embodiment whichrelates to the antenna arrays with bipolarization, in other words, thearrays which are made up of antenna elements polarized according to twodifferent polarizations.

For an antenna with bipolarization, the signal at the discrete Fouriertransform output is modeled as follows.

After spectral analysis, the signal measured in the time-frequency cellof index n is written, in the hypothesis H₁ (that is to say thehypothesis in which the signal is present in the cell):

$H_{1}\text{:}\mspace{14mu} \left\{ {{{\begin{matrix}{z_{1n} = {{g_{1} \cdot s_{n}} + w_{1n}}} \\{z_{2n} = {{g_{2} \cdot s_{n}} + w_{2n}}}\end{matrix}n} = 1},2,{\ldots \mspace{14mu} N}} \right.$

g₁ and g₂ are the complex gains of the two antennas, which depend on thepolarization.

The signal s_(n) is zero with the probability 1−q and different from 0with the probability q. s_(n) is a sample of a centered complex Gaussianvariable of variance 2σ′². The samples s_(n) are independent from onetime-frequency cell to another.

With 2σ′² being the total power of the useful signal, it is assumed,without loss of generality, that |g₁|²+|g₂|²=1 (usually g₁=cos(α) andg₂=sin(α)e_(iψ) are assumed).

w_(1n), w_(2n) are centered complex Gaussian noises of the same variance2σ², independent at n and mutually independent.

When the signal is absent, the following model H₀ applies:

$H_{0}\text{:}\mspace{14mu} \left\{ {{{\begin{matrix}{z_{1n} = w_{1n}} \\{z_{2n} = w_{2n}}\end{matrix}n} = 1},2,{\ldots \mspace{14mu} N}} \right.$

When s_(n) is different from 0, (z_(1n),z_(2n))=z_(n) ^(T) is a centeredcomplex Gaussian vector of covariance:

$R = \begin{pmatrix}{{{{g_{1}}^{2} \cdot 2}\sigma^{\prime 2}} + {2\sigma^{2}}} & {2g_{1}\overset{\_}{g_{2}}\sigma^{\prime 2}} \\{2\overset{\_}{g_{1}}g_{2}\sigma^{\prime 2}} & {{{g_{2}}^{2}2\sigma^{\prime 2}} + {2\sigma^{2}}}\end{pmatrix}$

Hereinbelow, G is used to denote the vector (g₁ g₂)^(T) such that

R=2σ′² GG*+2σ² I

For an antenna array comprising P antenna elements with bipolarization,the following models are deduced therefrom.

When the signal is present, the model H₁ applies:

$H_{1}\text{:}\mspace{14mu} \left\{ {{{\begin{matrix}{z_{1{pn}} = {{{g_{1} \cdot ^{{\phi}_{p}}}s_{n}} + w_{1{pn}}}} \\{z_{2{pn}} = {{{g_{2} \cdot ^{{\phi}_{p}}}s_{n}} + w_{2{pn}}}}\end{matrix}n} = 1},2,{\ldots \mspace{14mu} N}} \right.$

in which φ_(p) represents the geometrical phase shift of the antenna prelative to a point chosen as reference.

When the signal is absent, the model H₀ applies:

$H_{0}\text{:}\mspace{14mu} \left\{ {{{\begin{matrix}{z_{1{pn}} = w_{1{pn}}} \\{z_{2{pn}} = w_{2{pn}}}\end{matrix}n} = 1},2,{\ldots \mspace{14mu} N}} \right.$

The following notations are now used:

z_(p) _(n) the measurement n made on the antenna p: z_(p) _(n)=(z_(1pn),z_(2pn))^(T)

z_(n) all of the measurements of index n made on all of the P antennas:z=(z_(11n), z_(21n), . . . , z_(1pn), z_(2pn))^(T)

when s_(n)≠0, z_(p) _(n) is a centered complex Gaussian vector ofcovariance:

$R = \begin{pmatrix}{{{{g_{1}}^{2} \cdot 2}\sigma^{\prime 2}} + {2\sigma^{2}}} & {2g_{1}\overset{\_}{g_{2}}\sigma^{\prime 2}} \\{2\overset{\_}{g_{1}}g_{2}\sigma^{\prime 2}} & {{{g_{2}}^{2}2\sigma^{\prime 2}} + {2\sigma^{2}}}\end{pmatrix}$

For P antennas with bipolarization, if the measurements are assumedindependent from one antenna with bipolarization to another (whichamounts to making a good on average detector in all the directions ofarrival and abandoning making a detection optimized as a function of agiven direction of arrival), then, in the hypothesis H₁, the measurementvector (z_(11n), z_(21n), . . . , z_(1Pn), z_(2Pn))^(T) has thefollowing non-conditioned covariance matrix:

$\Re = \begin{pmatrix}R & \; & \; & \; & \; \\\; & R & \; & \; & \; \\\; & \; & \ldots & \; & \; \\\; & \; & \; & R & \; \\\; & \; & \; & \; & R\end{pmatrix}$

This is a diagonal block matrix; it therefore has the followingproperties:

det   ℜ = (det   R)^(P) $\Re^{- 1} = \begin{pmatrix}R^{- 1} & \; & \; & \; & \; \\\; & R^{- 1} & \; & \; & \; \\\; & \; & \ldots & \; & \; \\\; & \; & \; & R^{- 1} & \; \\\; & \; & \; & \; & R^{- 1}\end{pmatrix}$

There now follows a description of the detector implemented, accordingto the invention, for an antenna array comprising P elements withbipolarization. The invention applies notably for lacunary arrays withbipolarization in which the phase centers of the reception channels ofthe two polarizations of one and the same antenna coincide.

In the hypothesis H₀, the complex signal z has the probability density:

${p_{0}\left( z_{n} \right)} = {\frac{1}{{\pi^{2P}\left( {2\sigma^{2}} \right)}^{2P}}{\exp\left( {- \frac{{z_{n}}^{2}}{2\sigma^{2}}} \right)}}$

In the hypothesis H₁, the complex signal z_(n) has the probabilitydensity:

$\begin{matrix}{{p_{1}\left( z_{n} \right)} = {{\frac{q}{\pi^{2P}\mspace{11mu} \det \mspace{11mu} \Re}{\exp \left( {{- z_{n}^{*}}\Re^{- 1}z_{n}} \right)}} + {\frac{1 - q}{{\pi^{2P}\left( {2\sigma^{2}} \right)}^{2P}}{\exp\left( {- \frac{{z_{n}}^{2}}{2\sigma^{2}}} \right)}}}} \\{= {{\frac{q}{\pi^{2P}\mspace{11mu} \left( {\det \mspace{11mu} \Re} \right)^{P}}{\exp\left( {- {\sum\limits_{p = 1}^{P}{z_{pn}^{*}R^{- 1}z_{pn}}}} \right)}} + {\frac{1 - q}{{\pi^{2P}\left( {2\sigma^{2}} \right)}^{2P}}{\exp\left( {- \frac{{z_{n}}^{2}}{2\sigma^{2}}} \right)}}}}\end{matrix}$

After calculations, the following is obtained:

${R^{- 1} = {{\frac{1}{2\sigma^{2}}I} - {\frac{2\sigma^{\prime 2}{G}^{2}}{2{\sigma^{2}\left( {{2\sigma^{\prime 2}{G}^{2}} + {2\sigma^{2}}} \right)}}{\Gamma\Gamma}^{*}}}},{{{in}\mspace{14mu} {which}\mspace{14mu} \Gamma} = {G/{G}}}$

From the above equations, the likelihood ratio is deduced:

${L_{n:}L_{n}} = {\frac{p_{1}\left( z_{n} \right)}{p_{0}\left( z_{n} \right)} = {{\frac{{q\left( {2\sigma^{2}} \right)}^{2P}}{\left( {\det \mspace{11mu} R} \right)^{P}}{\exp\left( {\frac{2\sigma^{\prime 2}{G}^{2}}{2{\sigma^{2}\left( {{2\sigma^{\prime 2}{G}^{2}} + {2\sigma^{2}}} \right)}}{\sum\limits_{p = 1}^{P}{{\Gamma^{*}z_{pn}}}^{2}}} \right)}} + 1 - q}}$

The optimal detector in the sense of the Neyman-Pearson criterion (forthe values of q and σ′² chosen) is equivalent to comparing the functionln L(z₁, z₂, . . . z_(N); q, σ′², Γ)=Σ_(n=1) ^(N) ln p₁(z_(n))−lnp₀(z_(n)), to a threshold, with said function depending on three unknownparameters: Γ, σ′² and q, in which σ′² is the power of the usefulsignal, q is the proportion of useful signal within the window analyzed,and Γ is the eigenvector of R associated with the greatest eigenvalue ofR.

In effect R=2σ′² GG*+2σ²I, therefore: R·G=(2σ′²∥G∥²+2σ²)G,

G, and also Γ=G/∥G∥, are therefore eigenvectors of the matrix Rassociated with the eigenvalue 2σ′²∥G∥²+2σ².

The second eigenvalue of R has the value trR−(2σ′²∥G∥²+2σ²)=2σ².

Γ is therefore a eigenvector of R associated with the greatesteigenvalue of R.

In the case where q=1, L(z_(n); q, σ′², Γ) is simplified to:

${{L\left( {{z_{n};\sigma^{\prime 2}},\Gamma} \right)} = {\frac{\left( {2\sigma^{2}} \right)^{2P}}{\left( {\det \mspace{11mu} R} \right)^{P}}{\exp\left( {\frac{2\sigma^{\prime 2}{G}^{2}}{2{\sigma^{2}\left( {{2\sigma^{\prime 2}} + {2\sigma^{2}}} \right)}}{\sum\limits_{p = 1}^{P}{{\Gamma^{*}z_{pn}}}^{2}}} \right)}}},$

that can be put in the form:

$g_{\sigma,\sigma^{\prime}}\left( {\sum\limits_{p = 1}^{P}{{\Gamma^{*}z_{pn}}}^{2}} \right)$

in which g_(σ,σ′)(.) is the function defined by

${g_{\sigma,\sigma^{\prime}}(u)} = {\frac{\left( {2\sigma^{2}} \right)^{2P}}{\left( {\det \mspace{11mu} R} \right)^{P}}{\exp\left( \frac{2\sigma^{\prime 2}{G}^{2}u}{2{\sigma^{2}\left( {{2\sigma^{\prime 2}} + {2\sigma^{2}}} \right)}} \right)}}$

which is increasing monotone.

Consequently, the family p₁(z_(n)), indexed by σ′, is a function withmonotone likelihood ratio for the function

$\left( {\sum\limits_{p = 1}^{P}{{\Gamma^{*}z_{pn}}}^{2}} \right).$

Therefore the test

$\left( {\sum\limits_{p = 1}^{P}{{\Gamma^{*}z_{pn}}}^{2}} \right)$

above or below a threshold is a Uniformly Most Powerful (UMP) test,known in the field for example form the work “Testing StatisticalHypothesis”, E. L. Lehmann, J. P. Romano, Springer 2005. That meansthat, whatever the value of the unknown parameter a′, and for a setprobability of false alarm, the test has a probability of detection(power of the test) greater than that of any other test.

When there are N measurements available, the test becomes:

$\sum\limits_{n = 1}^{N}\left( {\sum\limits_{p = 1}^{P}{{\Gamma^{*}z_{pn}}}^{2}} \right)$

above or below a threshold.

Γ being unknown, it must be estimated.

Since q=1, it is possible to analytically calculate the estimate of Γ inthe sense of the likelihood maximum, {circumflex over (Γ)}_(MV).

{circumflex over (Γ)}_(MV) is the value of Γ which maximizes

$\prod\limits_{n = 1}^{N}{{L\left( z_{n} \right)}.}$

Now, when q=1, is

$\prod\limits_{n = 1}^{N}{L\left( z_{n} \right)}$

equal to

${\exp\left( {\sum\limits_{n = 1}^{N}{\sum\limits_{p = 1}^{P}{{\Gamma^{*}z_{n}}}^{2}}} \right)},$

to within a multiplying term close to

${\left( \frac{4\sigma^{4}}{\det \mspace{11mu} R} \right)^{P} = \left( \frac{2\sigma^{2}}{{2\sigma^{\prime 2}{G}^{2}} + {2\sigma^{2}}} \right)^{P}},$

which is independent of Γ because ∥G∥=1.

Therefore:

${{{{\prod\limits_{n = 1}^{N}{{L\left( z_{n} \right)}\mspace{14mu} {maximum}\mspace{14mu} \text{<=}\text{>}\mspace{14mu} {\sum\limits_{n = 1}^{N}{\sum\limits_{p = 1}^{P}{{{\Gamma^{*}z_{pn}}}^{2}\mspace{11mu} {maximum}\mspace{14mu} \text{<=}\text{>}}}}}}\mspace{14mu}\quad}{\sum\limits_{n = 1}^{N}{\sum\limits_{p = 1}^{P}{\Gamma^{*}z_{pn}z_{pn}^{*}\Gamma \mspace{11mu} {maximum}\mspace{14mu} \text{<=}\text{>}}}}}\mspace{14mu}\quad}{\Gamma^{*}\left( {\sum\limits_{n = 1}^{N}{\sum\limits_{p = 1}^{P}{z_{pn}z_{pn}^{*}}}} \right)}\Gamma \mspace{11mu} {maximum}$

<=>Γ*{circumflex over (R)}Γ maximum in which R is the empiricalcovariance matrix of the measurements.

This amounts to maximizing the quantity Z=Γ*{circumflex over(R)}Γ−λ(Γ*Γ−1), because Γ is normalized.

By deriving Z relative to Γ*, the following is obtained: ∂Z/∂Γ*=−λΓ=0.

The maximum of Z is therefore reached when F is taken equal to theeigenvector associated with the greatest eigenvalue of the empiricalcovariance matrix.

When q=1, {circumflex over (Γ)}_(MV) is therefore the eigenvectorassociated with the greatest eigenvalue of the empirical covariancematrix.

The test obtained by replacing Γ with {circumflex over (Γ)}_(MV) in

$\sum\limits_{n = 1}^{N}\left( {\sum\limits_{p = 1}^{P}{{\Gamma^{*}z_{pn}}}^{2}} \right)$

is the so-called GLRT (Generalized Likelihood Ratio Test) test known inthe field, for example via the work “Detection, Estimation andModulation” H. L. van Trees, Wiley 1968.

The GLRT method, which is often proposed when there is no uniformly morepowerful test, is not however applicable when the parameter q isdifferent from 1 because the likelihood ratio

$\prod\limits_{n = 1}^{N}{L\left( z_{n} \right)}$

is a law of mixture (sum of probability densities weighted by q and1−q), which makes its analytical resolution difficult, even impossible.

When q is different from 1, the invention proposes, as for themono-polarization arrays, optimizing the operation of the detector inthe vicinity of a signal-to-noise ratio and a coefficient q that arefixed.

For Γ, the same estimator is retained as previously and the maximumeigenvalue associated with the matrix

$\hat{R} = {\frac{1}{N}{\sum\limits_{n = 1}^{N}{\sum\limits_{p = 1}^{P}{z_{pn}z_{pn}^{*}}}}}$

is taken for {circumflex over (Γ)}.

This is justified by the relationship:

$\hat{R} = {{\frac{1}{N}{\sum\limits_{n = 1}^{N}{\sum\limits_{p = 1}^{P}{z_{pn}z_{pn}^{*}}}}} = {{qR} + {2\sigma^{2}I}}}$

to which are added terms centered at 1/√{square root over (N )}

The above relationship shows that {circumflex over (Γ)}′ is also aeigenvector of {circumflex over (R)} associated with its greatesteigenvalue: in effect, the matrix I has the eigenvalue 1 and accepts anyeigenvector, therefore any eigenvector of R (associated with theeigenvalue λ) is also a eigenvector of {circumflex over (R)} (associatedwith the eigenvalue qλ+2σ²).

This estimator is not the likelihood maximum, but it has the sameproperties as the likelihood maximum: it is not biased and itscovariance is 1/N.

Therefore, the structure of the Neyman-Pearson test

${\sum\limits_{n = 1}^{N}{\ln \mspace{11mu} {L\left( z_{n} \right)}}} > {or} < {threshold}$

is used by taking as the estimator of Γ the eigenvector of the empiricalcovariance matrix, associated with its greatest eigenvalue, and bychoosing a point of operation for the other parameters σ′² and q.

The performance levels of the detector are unchanged if the test ischanged to:

${\sum\limits_{n = 1}^{N}{T_{q}\left( z_{n} \right)}} = {{\sum\limits_{n = 1}^{N}\left( {{\alpha \mspace{11mu} \ln \mspace{11mu} {L\left( z_{n} \right)}} + \beta} \right)} > {or} < {threshold}}$

with α>0 and β real.

α and β are chosen such that the asymptotic behavior of the detector isthat of the quadratic detector, that is to say that

${{{\alpha \; {L\left( z_{n} \right)}} + \beta} \approx {\sum\limits_{p = 1}^{P}{{{{\Gamma^{*}z_{pn}}}^{2}/2}\sigma^{2}\mspace{14mu} {for}\mspace{14mu} {\sum\limits_{p = 1}^{P}{{\Gamma^{*}z_{pn}}}^{2}}}}}->{\infty.}$

Furthermore, it is desirable, for any q, for T_(q)(0)=0, which meansthat when the measured signal is zero, it is desirable for the criterionto be zero. These two conditions dictate:

$\quad\left\{ \begin{matrix}{{{\alpha \mspace{11mu} {\ln\left\lbrack {{q\left( \frac{4\sigma^{4}}{\det \mspace{11mu} R} \right)}^{P} + 1 - q} \right\rbrack}} + \beta} = 0} \\{{\alpha \frac{\; {2\sigma^{\prime 2}{G}^{2}}}{{2\sigma^{\prime 2}{G}^{2}} + {2\sigma^{2}}}} = 1}\end{matrix} \right.$

Hence, the following is obtained:

$\alpha = {\frac{{2\sigma^{\prime 2}{G}^{2}} + {2\sigma^{2}}}{2\sigma^{\prime 2}{G}^{2}}\mspace{14mu} {and}}$$\beta = {{- \frac{{2\sigma^{\prime 2}{G}^{2}} + {2\sigma^{2}}}{2\sigma^{\prime \; 2}{G}^{2}}}{\ln\left\lbrack {{q\left( \frac{4\sigma^{4}}{\det \mspace{11mu} R} \right)}^{- P} + 1 - q} \right\rbrack}\mspace{14mu} {now}}$G = 1${{Therefore}\text{:}\mspace{14mu} \alpha} = {\frac{{2\sigma^{\prime 2}} + {2\sigma^{2}}}{2\sigma^{\prime 2}}\mspace{14mu} {and}}$$\beta = {{- \frac{{2\sigma^{\prime 2}} + {2\sigma^{2}}}{2\sigma^{\prime 2}}}{\ln\left\lbrack {\frac{{q\left( {2\sigma^{2}} \right)}^{2P}}{\left( {\det \mspace{11mu} R} \right)^{P}} + 1 - q} \right\rbrack}}$

With the notations defined above, the characteristic of the detectorbecomes:

${T_{q}\left( Z_{n} \right)} = {\alpha \mspace{11mu} {\ln\left\lbrack \frac{{q\; {\gamma }^{Z_{n}/\alpha}} + 1 - q}{1 + {q\; \gamma} - q} \right\rbrack}\mspace{14mu} {with}}$$Z_{n} = {{\sum\limits_{p = 1}^{P}{{{{\Gamma^{*}z_{pn}}}^{2}/2}\sigma^{2}\mspace{14mu} {and}\mspace{14mu} \gamma}} = {\left( \frac{4\sigma^{4}}{\det \mspace{11mu} R} \right)^{P} = \frac{1}{\left( {1 + \frac{\sigma^{\prime 2}}{\sigma^{2}}} \right)^{P}}}}$

It is proposed to approach the characteristic of the detector by twostraight lines:

-   -   its asymptote at

${Z_{n}->{\infty:y}} = {x + {\alpha \mspace{11mu} {\ln \left( \frac{q\; \gamma}{{q\; \gamma} + 1 - q} \right)}}}$

-   -   the straight line: y=0 (the slope at the origin is not zero        therefore this straight line is not the asymptote of T_(q) at 0)

It therefore becomes:

$\sum\limits_{window}^{\;}{T_{q}\left( Z_{n} \right)}$ In  which:$\left\{ \begin{matrix}{{T_{q}\left( Z_{n} \right)} = 0} & {{{when}\mspace{14mu} Z_{n}} \leq s} \\{{T_{q}\left( Z_{n} \right)} = {Z_{n} - s}} & {{{when}\mspace{14mu} Z_{n}} \geq s}\end{matrix} \right.$

and in which s is defined by the point of intersection of the x-axiswith the asymptote at +∞.

There now follows a description, based on the flow diagram of FIG. 1, ofthe steps in implementing the various embodiments of the invention.

In a first step 101, common to all the embodiments of the invention, atime-frequency transform is applied to the signals received on eachantenna element of the array. More specifically, for each signal, anumber of discrete Fourier transforms are applied, temporallyoverlapped, in order to obtain a time-frequency representation of thesignal in the form of a grid of time-frequency cells each containing thespectrum of the signal for a given frequency interval and a given timeinterval.

There now follows a description of the sequence of the steps of themethod according to a first embodiment which relates to the antennaarrays with monopolarization.

x_(n,j) denotes the spectral value obtained for the time-frequency cellof index n measured for the antenna element of index j, hereinaftercalled “spectrum”. X_(n) denotes the vector made up of the spectra ofthe time-frequency cell of index n measured for all the antenna elementsof the array.

In a second step 103, the quadratic sum of the spectra x_(n,j) iscalculated, in other words the norm squared of the vector X_(n):

${X_{n}}^{2} = {{X_{n}^{*}X_{n}} = {\sum\limits_{j = 1}^{Psensors}{x_{n,j}^{*}x_{n,j}}}}$

or even the energy of the signal X_(n).

In a third step 104, the nonlinear function T_(q) is applied to theresult of the second step 103 in order to eliminate the time-frequencycells assumed to contain only noise.

$\quad\left\{ \begin{matrix}{{T_{q}\left( {X_{n}^{*}X_{n}} \right)} = 0} & {{{when}\mspace{14mu} X_{n}^{*}X_{n}} \leq s} \\{{T_{q}\left( {X_{n}^{*}X_{n}} \right)} = {{X_{n}^{*}X_{n}} - s}} & {{{when}\mspace{14mu} X_{n}^{*}X_{n}} \geq s}\end{matrix} \right.$

The threshold s is determined by searching for the intersection betweenthe x-axis and the asymptote at infinity of the logarithm of thelikelihood ratio approximated by the function T_(q) given by therelationship (1). An expression of the value of the threshold s is givenby the relationship (2).

This relationship depends on the signal-to-noise ratio and on theparameter q representative of the probability of presence of the signalin a time-frequency cell.

To calculate the threshold s, the signal-to-noise ratio is set at agiven operating point, dependent notably on the targeted application.

For a signal-to-noise ratio stronger than the set operating point, thedetector is mismatched but this does not pose any problem because thesignal is more easy to detect than at the operating point for which thealgorithm is set. For a signal-to-noise ratio lower than the setoperating point, the detector is mismatched but it is considered thatthe corresponding signals are not of interest.

The value of the parameter q is set as close as possible to the value 0,without being zero, so as to obtain the case where the signal has a lowprobability of being present. The detector according to the invention isoptimized for this value of q. If the real value of q is greater thanthe value of q chosen, the average signal-to-noise ratio qσ′²/σ² ishigher and therefore the detection threshold will be more easilycrossed. If the real value of q is lower than the value of q chosen, theaverage signal-to-noise ratio is lower but it is considered that thissituation is of no interest. In practice, a value of q equal to 0.1 canbe retained because, below this value, the probability of presence ofthe signal is too low to constitute a realistic case.

In a fourth step 105, the result of the preceding step 104 is integratedover all of the time-frequency cells of the selected window.

Finally, in a last step 106, the result of the integration 105 iscompared to a predetermined detection threshold S_(det) to deducetherefrom the presence or the absence of signal.

The detection threshold S_(det) is determined by trial and error so asto set a given probability of false alarm.

In a second embodiment, the detection method according to the inventioncan also be applied for an antenna array with bipolarization.

In this case, the method according to the invention comprises anadditional step 102 which consists in calculating the empiricalcovariance matrix 2×2 over the P×N measurements of a window in which Pis the number of antenna elements with bipolarization and N the numberof time-frequency cells of the window:

$\hat{R} = \begin{pmatrix}{\sum\limits_{n = 1}^{N}{\sum\limits_{p = 1}^{P}{z_{1{pn}}^{*}z_{1{pn}}}}} & {\sum\limits_{n = 1}^{N}{\sum\limits_{p = 1}^{P}{z_{1{pn}}^{*}z_{2{pn}}}}} \\{\sum\limits_{n = 1}^{N}{\sum\limits_{p = 1}^{P}{z_{2{pn}}^{*}z_{1{pn}}}}} & {\sum\limits_{n = 1}^{N}{\sum\limits_{p = 1}^{P}{z_{2{pn}}^{*}z_{2{pn}}}}}\end{pmatrix}$

Then, the eigenvector {circumflex over (Γ)} associated with the greatesteigenvalue of {circumflex over (R)} is determined. The eigenvector{circumflex over (Γ)} can be determined, for example, by firstdiagonalizing the matrix {circumflex over (R)}.

The step 103 is then replaced by the calculation, for eachtime-frequency cell of index n, of the sum

$Z_{n} = {\sum\limits_{p = 1}^{P}{{{{\hat{\Gamma}}^{*}z_{pn}}}^{2}.}}$

This amounts to forming the channel on which the signal-to-noise ratiois maximum.

The following steps 104,105,106 are applied in the same way as for thecase of the mono-polarization array by replacing the power of the signalX_(n) with the sum Z_(n).

According to a variant of the second embodiment described above, onepossible line for simplification consists in considering that thecovariance matrix R is a diagonal matrix. This amounts to assuming thatthere is incoherence in the signals received, not only from one antennaelement to another, but also between the two channels, polarizeddifferently, of one and the same antenna element. The detector accordingto the invention then becomes quadratic in polarization and on thereception channels. This variant is applicable whether the phase centersof the two sub-arrays are collocated or not.

In this case, the step 102 becomes optional and the method is identicalto the case of the array with mono-polarization by replacing, in thestep 103, the power of the signal X_(n) with that of the signal made upof the spectra z_(pn) for a given time-frequency cell

$Z_{n} = {\sum\limits_{p = 1}^{P}{{z_{pn}}^{2}.}}$

This variant is of interest notably when the power of the receivedsignal is similar for the two polarizations. In this case, the inventionmakes it possible to transform the incoherent integration gain over theP reception channels of an array with mono-polarization into anincoherent integration gain over the 2P channels of the array withbipolarization.

One advantage of this variant is that it offers greater ease ofimplementation at the cost of an acceptable degradation of thesensitivity performance levels due to the fact that the integration gainin polarization is no longer a coherent integration gain.

FIGS. 2a, 2b, 3a, 3b schematically represent, on a number of blockdiagrams, the device for detecting electromagnetic signals according toa number of embodiments of the invention.

FIG. 2a describes a detection device according to a first embodiment ofthe invention applied to an antenna array with mono-polarization.

The device 200 described in FIG. 2a comprises an antenna array made upof a plurality of antenna elements or sensors A₁, A₂, A₃ . . . A_(P)with mono-polarization. Each antenna element is coupled to a receptionchannel R₁, R₂, R₃ . . . R_(P) to, notably, digitize the analog signalreceived by each sensor. At the output of each reception channel, atime-frequency transform DFT₁, DFT₂, DFT₃, DFT_(P) is applied, throughone or more temporally overlapped discrete Fourier transforms. For eachreception channel, this operation culminates in the construction of agrid of time-frequency cells each containing the spectrum of the signalfor a given time interval and a given frequency interval.

The detection device 200 also comprises a first computation module 201to perform the quadratic sum, in each case, of the spectra at the outputof each time-frequency transform. In other words, the computation module201 is configured to execute the step 103 of the detection methodaccording to the invention.

The detection device 200 also comprises a second computation module 202configured to apply the step of nonlinearity 104 of the method accordingto the invention, a third computation module 203 to perform anintegration, in accordance with the step 105 of the method according tothe invention, of the output of the second module 202 over atime-frequency window [ΔT_(k), Δf_(m)] which comprises a given number oftime-frequency cells. Finally, a last computation module 204 isconfigured to compare the result of the integration carried out by thethird module 203 to a predetermined detection threshold and produceinformation concerning the presence or the absence of signal in thetime-frequency window [ΔT_(k), Δf_(m)].

FIG. 2b schematically represents a variant of the detection device ofFIG. 2a according to which, for each reception channel R₁, R₂, R₃ . . .R_(P), two types of discrete Fourier transforms are applied with twodifferent frequency resolutions. In this case, the computation modules201-204 described for FIG. 2a are duplicated for each frequencyresolution. One advantage of having a number of types of discreteFourier transforms is that this makes it possible to increase theprobability of there being a filter width matched to the band of thesignal to be processed.

FIG. 3a represents a block diagram of a detection device 300 accordingto the invention applicable for an antenna array with bipolarization.

Such a device 300 comprises a plurality of antenna elements suitable foroperating according to two distinct polarizations. For a given channel,an antenna element with bipolarization A₁,A′₁ can be made up of twodistinct elements or of one single element configured to operateaccording to two distinct polarizations.

The device 300 comprises a reception channel Rp, R′p and atime-frequency transform DFT₁, DFT′₁, . . . DFT_(P), DFT_(P) for eachantenna element and each polarization.

The device 300 also comprises a number of computation modules301,302,303,304 configured to execute the steps 102 to 106 ofimplementation of the method for detecting electromagnetic signalsaccording to the invention.

Just as for the device with mono-polarization described in FIG. 2b , thedevice with bipolarization can also implement, for each receptionchannel R₁, R₂, R₃ . . . R_(P), two types of discrete Fourier transformswith two different frequency resolutions.

FIG. 3b represents a variant embodiment of the device of FIG. 3a ,according to which each polarization is processed separately at theoutput of the time-frequency transforms. A first computation module 311is configured to execute the step 103 of the method according to theinvention for the signals received according to a first polarization. Asecond computation module 321 is configured to execute the step 103 ofthe method according to the invention for the signals received accordingto a second polarization.

A third computation module 312 is configured to execute the step 104 ofnonlinearity of the method according to the invention for the signalsreceived according to a first polarization. A fourth computation module322 is configured to execute the step 104 of nonlinearity of the methodaccording to the invention for the signals received according to asecond polarization.

Two distinct modules 313, 323 perform the integration of the outputs ofthe modules 312, 322 over a given time-frequency window. An additionalmodule 314 is used to compare the outputs of the two integrators 313,323 and retain the output which exhibits the highest value. This lattervalue is compared to the detection threshold via a comparison module315.

The variant embodiment represented in FIG. 3b is notably advantageouswhen the power of the signal is not balanced over the two sub-arrayseach made up of antenna elements configured according to a givenpolarization.

This variant amounts to applying the method according to the invention,as described for the case of an array with mono-polarization, to each ofthe two sub-arrays operating in mono-polarization mode then to retainingonly the maximum of the detection results supplied over the twosub-arrays. This variant is applicable whether the phase centers of thetwo sub-arrays are collocated or not.

In the different variant embodiments of the device for detectingelectromagnetic signals according to the invention, the computationmodules can be organized according to different architectures, inparticular each step of the method can be implemented by a distinctmodule or, on the contrary, all of the steps can be combined within asingle computation module.

Each of the computation modules that the device according to theinvention comprises can be produced in software and/or hardware form.Each module can notably consist of a processor and a memory. Theprocessor can be a generic processor, a specific processor, anapplication-specific integrated circuit (also known by the acronym ASIC)or a field-programmable gate array (also known by the acronym FPGA).

1. A method for detecting an electromagnetic signal, called signal ofinterest, likely to be contained in an electromagnetic signal receivedby an antenna array comprising a plurality of antenna elements, saidmethod comprising the following steps: applying to said electromagneticsignal received by each antenna element a plurality of time-frequencytransforms in order to obtain a representation of said signal in theform of a plurality of time-frequency cells each containing the spectrumof said signal for a given frequency interval and a given time interval,for each time/frequency cell of a given set of cells, calculating theenergy of the vector made up of the spectra over all of the antennaelements, applying the following nonlinear function T to the result ofthe preceding energy calculation, so as to cancel the time/frequencycells containing substantially only noise: if the norm of the energy ofthe vector of the spectra is below a first predetermined threshold s,the result of the function T is zero, if the norm of the energy of thevector of the spectra is above or equal to said first threshold s, theresult of the function T is equal to the norm of the energy of thevector of the spectra minus the value of said first threshold s,integrating, over said set of time/frequency cells, the result of saidnonlinear function T, comparing the result of the integration to asecond predetermined threshold, called detection threshold S_(det), todetect the presence of the signal of interest.
 2. The method fordetecting an electromagnetic signal of claim 1, wherein said antennaelements are mono-polarized.
 3. The method for detecting anelectromagnetic signal of claim 1, wherein said antenna elements arebipolarized.
 4. The method for detecting an electromagnetic signal a ofclaim 3, wherein the energy calculation step is performed on the vectormade up of the spectra for each of the polarizations over all of theantenna elements.
 5. The method for detecting an electromagnetic signalof claim 4, also comprising: calculating the covariance matrix betweenthe plurality of spectra associated with the plurality of antennaelements configured according to a first polarization and the pluralityof spectra associated with the plurality of antenna elements configuredaccording to a second polarization, calculating the eigenvectorassociated with the greatest eigenvalue of said covariance matrix, inthe energy calculation step, replacing the spectrum with its scalarproduct with said eigenvector.
 6. The method for detecting anelectromagnetic signal of claim 3, wherein the steps of calculation ofthe energy of the spectra, of application of the nonlinear function andof integration, over a plurality of time/frequency cells, of the resultof said nonlinear function T, are performed separately for eachpolarization of said antenna elements, said method further comprising anadditional step of determination of the maximum of the results ofintegration over the two polarizations, said maximum being compared tosaid second detection threshold.
 7. The method for detecting anelectromagnetic signal of claim 1, wherein said first threshold s isdetermined by searching for the intersection between the x-axis and theasymptote at infinity of the logarithm of the likelihood ratio definedas the quotient of the probability densities in the hypothesis in whichthe signal of interest is present in a time/frequency cell and in thehypotheses (H₀) in which the signal of interest is absent in atime/frequency cell.
 8. The method for detecting an electromagneticsignal of claim 7, wherein said first threshold s is calculated as afunction of a given signal-to-noise ratio and of a parameter qrepresentative of the probability of presence of the signal of interestin a time/frequency cell.
 9. The method for detecting an electromagneticsignal of claim 8, wherein said parameter q is chosen from a range lyingbetween 0.1 and
 1. 10. The method for detecting an electromagneticsignal of claim 1, wherein said second detection threshold S_(det) isconfigured to observe a given probability of false alarm.
 11. The methodfor detecting an electromagnetic signal of claim 1, wherein said antennaarray is lacunary.
 12. A device for the detection of an electromagneticsignal, called signal of interest, likely to be contained in anelectromagnetic signal, said device comprising an antenna array formedby a plurality of antenna elements and means configured to implement themethod of claim 1.